Floquet systems, which could be characterized by periodically modulated Hamiltonian, display rich dynamics and novel phenomena that are absent in their un-modulated counterparts, such as quantum dynamical decoupling [1]}, [2]}, [3]}, time crystal [4]}, [5]}, Mach-Zehnder
interferometer [6]}, [7]}, time-domain Fresnel lenses [8]}, time-domain grating [9]} and Floquet topological phase [10]}. The studies of periodically modulated systems are also known as Floquet engineering  [11]}, [12]}, [13]}, [14]}, [15]}. Moreover, for the Floquet systems, the dynamic steady-states are periodic steady-states, which emerge in a balance of the energy injection by the periodic driving and the relaxation processes [16]}, [17]}, [18]}. To explore the stability properties of the periodic steady-states, the time-average values of observable physical quantities are usually observed experimentally [19]}, [9]}, and stroboscopic evolution of a periodically driven quantum system in steps of the modulation period is usually adopted in theoretical studies [21]}.
